Uncertainty quantification and sensitivity analysis of left ventricular function during the full cardiac cycle.

Patient-specific computer simulations can be a powerful tool in clinical applications, helping in diagnostics and the development of new treatments. However, its practical use depends on the reliability of the models. The construction of cardiac simulations involves several steps with inherent uncertainties, including model parameters, the generation of personalized geometry and fibre orientation assignment, which are semi-manual processes subject to errors. Thus, it is important to quantify how these uncertainties impact model predictions. The present work performs uncertainty quantification and sensitivity analyses to assess the variability in important quantities of interest (QoI). Clinical quantities are analysed in terms of overall variability and to identify which parameters are the major contributors. The analyses are performed for simulations of the left ventricle function during the entire cardiac cycle. Uncertainties are incorporated in several model parameters, including regional wall thickness, fibre orientation, passive material parameters, active stress and the circulatory model. The results show that the QoI are very sensitive to active stress, wall thickness and fibre direction, where ejection fraction and ventricular torsion are the most impacted outputs. Thus, to improve the precision of models of cardiac mechanics, new methods should be considered to decrease uncertainties associated with geometrical reconstruction, estimation of active stress and of fibre orientation. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Sensitivity of stress and strain calculations to passive material parameters in cardiac mechanical models using unloaded geometries.

Cardiac stress (load) and strain (stretch) are widely studied indicators of cardiac function and outcome, but are difficult or impossible to directly measure in relation to the cardiac microstructure. An alternative approach is to estimate these states using computer methods and image-based measurements, but this still requires knowledge of the tissue material properties and the unloaded state, both of which are difficult to determine. In this work, we tested the sensitivity of these two interdependent unknowns (reference geometry and material parameters) on stress and strain calculations in cardiac tissue. Our study used a finite element model of the human ventricle, with a hyperelastic passive material model, and was driven by a cell model mediated active contraction. We evaluated 21 different published parameter sets for the five parameters of the passive material model, and for each set we optimised the corresponding unloaded geometry and contractility parameter to model a single pressure-volume loop. The resulting mechanics were compared, and calculated systolic stresses were largely insensitive to the chosen parameter set when an unloading algorithm was used. Meanwhile, material strain calculations varied substantially depending on the choice of material parameters. These results indicate that determining the correct material and unloaded configuration may be highly important to understand strain driven processes, but less so for calculating stress estimates.

Space-discretization error analysis and stabilization schemes for conduction velocity in cardiac electrophysiology.

In cardiac electrophysiology, the propagation of the action potential may be described by a set of reaction-diffusion equations known as the bidomain model. The shape of the solution is determined by a balance of a strong reaction and a relatively weak diffusion, which leads to steep variations in space and time. From a numerical point of view, the sharp spatial gradients may be seen as particularly problematic, because computational grid resolution on the order of 0.1 mm or less is required, yielding considerable computational efforts on human geometries. In this paper, we discuss a number of well-known numerical schemes for the bidomain equation and show how the quality of the solution is affected by the spatial discretization. In particular, we study in detail the effect of discretization on the conduction velocity (CV), which is an important quantity from a physiological point of view. We show that commonly applied finite element techniques tend to overestimate the CV on coarse grids, while it tends to be underestimated by finite difference schemes. Furthermore, the choice of interpolation and discretization scheme for the nonlinear reaction term has a strong impact on the CV. Finally, we exploit the results of the error analysis to propose improved numerical methods, including a stabilized scheme that tends to correct the CV on coarse grids but converges to the correct solution as the grid is refined. Copyright © 2016 John Wiley & Sons, Ltd.

Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations.

Mathematical models of cardiac electro-mechanics typically consist of three tightly coupled parts: systems of ordinary differential equations describing electro-chemical reactions and cross-bridge dynamics in the muscle cells, a system of partial differential equations modelling the propagation of the electrical activation through the tissue and a nonlinear elasticity problem describing the mechanical deformations of the heart muscle. The complexity of the mathematical model motivates numerical methods based on operator splitting, but simple explicit splitting schemes have been shown to give severe stability problems for realistic models of cardiac electro-mechanical coupling. The stability may be improved by adopting semi-implicit schemes, but these give rise to challenges in updating and linearising the active tension. In this paper we present an operator splitting framework for strongly coupled electro-mechanical simulations and discuss alternative strategies for updating and linearising the active stress component. Numerical experiments demonstrate considerable performance increases from an update method based on a generalised Rush-Larsen scheme and a consistent linearisation of active stress based on the first elasticity tensor.

Effects of deformation on transmural dispersion of repolarization using in silico models of human left ventricular wedge.

Mechanical deformation affects the electrical activity of the heart through multiple feedback loops. The purpose of this work is to study the effect of deformation on transmural dispersion of repolarization and on surface electrograms using an in silico human ventricular wedge. To achieve this purpose, we developed a strongly coupled electromechanical cell model by coupling a human left ventricle electrophysiology model and an active contraction model reparameterized for human cells. This model was then embedded in tissue simulations on the basis of bidomain equations and nonlinear solid mechanics. The coupled model was used to evaluate effects of mechanical deformation on important features of repolarization and electrograms. Our results indicate an increase in the T-wave amplitude of the surface electrograms in simulations that account for the effects of cardiac deformation. This increased T-wave amplitude can be explained by changes to the coupling between neighboring myocytes, also known as electrotonic effect. The thickening of the ventricular wall during repolarization contributes to the decoupling of cells in the transmural direction, enhancing action potential heterogeneity and increasing both transmural repolarization dispersion and T-wave amplitude of surface electrograms. The simulations suggest that a considerable percentage of the T-wave amplitude (15%) may be related to cardiac deformation.

Uncertainty analysis of ventricular mechanics using the probabilistic collocation method.

Uncertainty and variability in material parameters are fundamental challenges in computational biomechanics. Analyzing and quantifying the resulting uncertainty in computed results with parameter sweeps or Monte Carlo methods has become very computationally demanding. In this paper, we consider a stochastic method named the probabilistic collocation method, and investigate its applicability for uncertainty analysis in computing the passive mechanical behavior of the left ventricle. Specifically, we study the effect of uncertainties in material input parameters upon response properties such as the increase in cavity volume, the elongation of the ventricle, the increase in inner radius, the decrease in wall thickness, and the rotation at apex. The numerical simulations conducted herein indicate that the method is well suited for the problem of consideration, and is far more efficient than the Monte Carlo simulation method for obtaining a detailed uncertainty quantification. The numerical experiments also give interesting indications on which material parameters are most critical for accurately determining various global responses.

Numerical solution of the bidomain equations.

Knowledge of cardiac electrophysiology is efficiently formulated in terms of mathematical models. However, most of these models are very complex and thus defeat direct mathematical reasoning founded on classical and analytical considerations. This is particularly so for the celebrated bidomain model that was developed almost 40 years ago for the concurrent analysis of extra- and intracellular electrical activity. Numerical simulations based on this model represent an indispensable tool for studying electrophysiology. However, complex mathematical models, steep gradients in the solutions and complicated geometries lead to extremely challenging computational problems. The greatest achievement in scientific computing over the past 50 years has been to enable the solving of linear systems of algebraic equations that arise from discretizations of partial differential equations in an optimal manner, i.e. such that the central processing unit (CPU) effort increases linearly with the number of computational nodes. Over the past decade, such optimal methods have been introduced in the simulation of electrophysiology. This development, together with the development of affordable parallel computers, has enabled the solution of the bidomain model combined with accurate cellular models, on geometries resembling a human heart. However, in spite of recent progress, the full potential of modern computational methods has yet to be exploited for the solution of the bidomain model. This paper reviews the development of numerical methods for solving the bidomain model. However, the field is huge and we thus restrict our focus to developments that have been made since the year 2000.

Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart.

The electrical activity of the heart may be modeled with a system of partial differential equations (PDEs) known as the bidomain model. Computer simulations based on these equations may become a helpful tool to understand the relationship between changes in the electrical field and various heart diseases. Because of the rapid variations in the electrical field, sufficiently accurate simulations require a fine-scale discretization of the equations. For realistic geometries this leads to a large number of grid points and consequently large linear systems to be solved for each time step. In this paper, we present a fully coupled discretization of the bidomain model, leading to a block structured linear system. We take advantage of the block structure to construct an efficient preconditioner for the linear system, by combining multigrid with an operator splitting technique.

Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells.

The contraction of the heart is preceded and caused by a cellular electro-chemical reaction, causing an electrical field to be generated. Performing realistic computer simulations of this process involves solving a set of partial differential equations, as well as a large number of ordinary differential equations (ODEs) characterizing the reactive behavior of the cardiac tissue. Experiments have shown that the solution of the ODEs contribute significantly to the total work of a simulation, and there is thus a strong need to utilize efficient solution methods for this part of the problem. This paper presents how an efficient implicit Runge-Kutta method may be adapted to solve a complicated cardiac cell model consisting of 31 ODEs, and how this solver may be coupled to a set of PDE solvers to provide complete simulations of the electrical activity.